I don't actually believe the Adam and Eve passages to be real but rather a construct for a paradigm on the origins of life (I tend to accept the scientific explanation behind the origins of man).
I agree with you on this. When I read the Adam and Eve story (and some of the other very early Genesis stories), I read it as a story written by the ancient Hebrews to convey important truths about human nature and humans' relationship to God. So, mythological, in the sense of a fictional story intended to convey deep and important ideas.
Right now, I don't know why the story's author has Eve taking the fruit first, then Adam. I'm not satisfied with explanations that involve hierarchy or authority, because I don't see that elsewhere in the story. When woman is created, the storyteller goes to some length to say that the woman is like the man, in a way that the animals can never be. There's equality in that part of the story. We see "he shall rule over thee" at the end, but it's part of the curse: because humans sinned, relationships are broken in several ways, and this is one of the ways.
I will say that, with the story arranged as it is, it allows the delightful cascade of blaming at the end: "Eve made me do it!" "The serpent made me do it!" Very human, that.
On misogyny more generally in the Scriptures: I see both in the Bible, women having terribly low status in many portions of the Bible, but also strong affirmation of the full worth of women. I'm hoping we can work through this mixture of biblical viewpoints as this discussion progresses.